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TL;DR

The problem of choosing the best time to take a particular action based on sequentially observed information. Formally, find the stopping time \( \tau \) that maximizes \( E[g(X_\tau)] \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Optimal Stopping

The problem of choosing the best time to take a particular action based on sequentially observed information. Formally, find the stopping time \( \tau \) that maximizes \( E[g(X_\tau)] \).

Why it matters for interviews

Optimal stopping encompasses American option exercise, the secretary problem, and many trading decisions (when to enter/exit positions). It combines probability, dynamic programming, and decision theory.

Definition and Mathematical Foundation

The problem of choosing the best time to take a particular action based on sequentially observed information. Formally, find the stopping time \( \tau \) that maximizes \( E[g(X_\tau)] \).

Application in Quantitative Finance

Optimal stopping encompasses American option exercise, the secretary problem, and many trading decisions (when to enter/exit positions). It combines probability, dynamic programming, and decision theory.

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Frequently Asked Questions

What is the connection between optimal stopping and free boundary problems?▼
In continuous time, optimal stopping produces a free boundary separating the 'stop' and 'continue' regions. For American options, this is the early exercise boundary. The value function satisfies a variational inequality.
What is the optimal stopping rule for the secretary problem?▼
Observe and reject the first \( n/e \) candidates, then hire the next one who is better than all previously seen. This achieves a success probability of \( 1/e \approx 36.8\% \), which is optimal.
How is optimal stopping solved numerically?▼
Backward induction on a discretized state space, or regression-based methods (Longstaff-Schwartz for American options). At each step, compare the exercise value with the continuation value estimated from regression.